Thermodynamics cooling

Ok hi guys,
recently i was doing a practical on thermodynamics. Then i came across of a question which was to comment on the assumption that the rate of heat loss for 2 different liquids , placed in identical calorimetry copper cups was the same.( the 2 liquids started off at the same temperature ( the liquids were oil and water)).

Ok my reasoning for the rate of heat loss being the same was that the copper cup,can only have a loss in energy at a fixed amount of power as the dimensions of the cup are kept constant. Please correct me if i am wrong.

Then i also commented on newtons law of cooling( rate of change of temperature is proportional to the difference in temperature between the objects temperature and the ambient temperature) If it were to be written in a equation,it would be the (rate of change in temperature = -k(T - Troom) . Okay the k here is a constant which is different for every object.My reasoning is that to bring down the temperature of 2 different objects by a kelvin would require different amounts of energy as the the specific heat capacity of every object is unique. Furthermore the rate of energy loss is the same,hence 2 different objects would not have the same k value.As they would require different amounts of time to cool down. Please comment on these statements

well buyer beware as its free advice and i have never taken thermodynamics course.
IME its usually most fruitful to compare the most polar opposites possible, even where it may make little practical sense.

So into cup 1 I place water, well characterized, and the basis for the calorie unit.

Into cup 2, I pour the same volume from a neutron star, magically cooled to the same temperature but just as dense. Or a cup of magma.

Will the rate of heat loss vary? In the first instant, i should think not. In the second, maybe. By the twentieth instant, most assuredly. There is still a significant gradient with the container of enormous heat capacity, whereas with water there may not be. Thats why the earths core remains molten while we scrub our backs with pumice stone. So in essence I agree, very different rates of cooling and heat flux thru the container as a fx of time.